Multi-Level Algebraic Systems: A Study of Hyperfields and n-Superhyperfields

Mathematical structures can be extended to hyperstructures and n-superhyperstructures by leveraging the powerset and its n-fold iteration (cf. Smarandache (2023)). These generalized frameworks are particularly well suited for modeling hierarchical relationships across diverse domains. In this paper, we examine the hyperfield and superhyperfield introduced in Fujita (2025). Recall that a field is an algebraic structure (F, +, ·) in which (F, +) and (F \ {0}, ·) are commutative groups and multiplication distributes over addition. A hyperfield generalizes this by replacing addition with a hyperaddition that maps each pair of elements to a subset of F, while preserving a commutative multiplicative group and distributivity. A superhyperfield further extends a hyperfield by lifting its operations to iterated powersets, yielding multi-level hyperaddition and multiplication structures.